A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.

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